[2] Four days later, during their first ever appearance in the UEFA Europa League, he netted twice in a 2–1 success at Omonia as his team played 30 minutes with one player less and qualified to the third round;[3] due to his exploits, he was dubbed the "Hero of Nicosia".

[4] On 21 August 2014, Budescu converted a penalty against French side Lyon in the first leg of the Europa League play-off (2–1 away win, qualification on the away goals rule).

[5] On 6 August of the following year, in the same competition but in the third qualifying round, his second leg brace helped defeat West Ham United 2–1 at home[6] and 4–3 on aggregate.

[7] The following month, he signed a three-year contract with China League One club Dalian Yifang for an undisclosed fee, with a salary worth €3 million plus €500,000 upon signature.

[17] He again found the net nine days later, converting a free kick in 2–2 home draw with Viktoria Plzeň in the Champions League third qualifying round.

[20] On 28 June 2018, Budescu agreed to a three-year deal with Saudi Professional League team Al-Shabab for a reported transfer fee of €2.5 million,[21][22] rejoining his former Astra Giurgiu head coach Marius Șumudică.

On 19 August 2019, Astra Giurgiu owner Ioan Niculae confirmed that Budescu returned for a fourth stint at the club after agreeing to a two-year contract.

After recovering from a long-term injury, Budescu registered his first appearance and goal in the top flight for Petrolul on 24 October, converting a penalty in a 2–0 home win over Argeș Pitești.

[35][36] On 11 October 2015, Budescu scored a brace against the Faroe Islands, helping the country achieve qualification by finishing second in the group one point behind Northern Ireland.

[43] In October 2014, Budescu married his partner Mădălina in the city of Ploiești, with then Astra Giurgiu teammate Daniel Coman acting as best man.